Global solutions in the critical Besov space for the non cutoff Boltzmann equation

TL;DR

This paper proves the existence of unique global solutions to the non-cutoff Boltzmann equation in a critical Besov space, advancing understanding of kinetic equations without angular cutoff assumptions.

Contribution

It introduces a novel analytical framework combining Chemin-Lerner and non-isotropic norms to establish global solutions in a critical function space for the non-cutoff Boltzmann equation.

Findings

Existence of unique global solutions in a critical Besov space.

Solution non-negativity is rigorously proven.

Method extends analytical tools for non-cutoff kinetic equations.

Abstract

The Boltzmann equation is studied without the cutoff assumption. Under a perturbative setting, a unique global solution of the Cauchy problem of the equation is established in a critical Chemin-Lerner space. In order to analyse the collisional term of the equation, a Chemin-Lerner norm is combined with a non-isotropic norm with respect to a velocity variable, which yields an apriori estimate for an energy estimate. Together with local existence following from commutator estimates and the Hahn-Banach extension theorem, the desired solution is obtained. Also, the non-negativity of the solution is rigorously shown.